Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#13286 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C2×D8 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C22×C7 | C2×#102 |
| prime radical | C23×C7 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ C2×C4×C7×C8 ◁ G | |
342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 47 points:
(1,2,3,4,5,6,7)(8,9)(10,14)(11,12)(13,15)(16,17,22,27,23,28,21,39)(18,29,33,41,34,46,32,44)(19,26,20,38,37,40,36,25)(24,30,42,47,43,35,45,31);(9,12)(13,14)(16,18,20,31,22,33,37,30,23,34,36,47,21,32,19,35)(17,24,40,46,39,45,38,41,28,43,26,29,27,42,25,44);(8,10,11,15)(9,13,12,14)(16,19,21,36,23,37,22,20)(17,25,27,26,28,38,39,40)(18,30,32,31,34,35,33,47)(24,41,42,46,43,44,45,29)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6274);